|
|
A343019
|
|
a(n) is the smallest number m such that tau(m+1) = tau(m) - n.
|
|
3
|
|
|
2, 4, 6, 16, 12, 24, 30, 36, 84, 324, 60, 144, 192, 120, 210, 288, 180, 528, 240, 576, 480, 360, 420, 900, 1344, 960, 720, 5184, 1008, 840, 1320, 2400, 1260, 17424, 1800, 14640, 2640, 1680, 2160, 8280, 4800, 3600, 11220, 7056, 3780, 6240, 2520, 82944, 6480
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
tau(m) = the number of divisors of m (A000005).
Sequences of numbers m such that tau(m+1) = tau(m) - n for 0 <= n <= 5:
n = 0: 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, ... (A005237).
n = 1: 4, 8, 81, 441, 625, 1089, 2024, 2401, 3025, 3968, ... (A068208).
n = 2: 6, 10, 20, 22, 32, 45, 46, 50, 58, 68, 76, 82, 92, ... (A227874).
n = 3: 16, 64, 224, 675, 1444, 2115, 3843, 5475, 6724, 9801, ...
n = 4: 12, 18, 28, 52, 54, 56, 105, 110, 114, 128, 148, 154, ...
n = 5: 24, 80, 225, 484, 1024, 1088, 1156, 1225, 1521, 2116, ...
|
|
LINKS
|
|
|
EXAMPLE
|
For n = 3; a(3) = 16 because 16 is the smallest number such that tau(17) = 2 = tau(16) - 3 = 5 - 3.
|
|
MATHEMATICA
|
d = Differences @ Table[DivisorSigma[0, n], {n, 1, 10^5}]; a[n_] := If[(p = Position[d, -n]) != {}, p[[1, 1]], 0]; s = {}; n = 0; While[(a1 = a[n]) > 0, AppendTo[s, a1]; n++]; s (* Amiram Eldar, Apr 03 2021 *)
|
|
PROG
|
(Magma) Ax:=func<n|exists(r){m: m in[1..10^6] | #Divisors(m + 1) - #Divisors(m) eq -n} select r else 0>; [Ax(n): n in [0..50]]
(PARI) a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) - n, m++); m; \\ Michel Marcus, Apr 03 2021
(Python)
from itertools import count, pairwise
from sympy import divisor_count
def A343019(n): return next(m+1 for m, t in enumerate(pairwise(map(divisor_count, count(1)))) if t[1] == t[0]-n) # Chai Wah Wu, Jul 25 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|